Is there a key combo/way to enable fine control mode when dragging automation points?
I don’t think so (hope I am wrong). The “fineness” is related to the horizontal and vertcal zoom. Meaning the more zoom you have the finer you can adjust the automation.
Regards. 
That’s disappointing. I don’t want to have to expand the automation track every time I want to adjust it.
Agreed.
Regards.